%0 Journal Article
%A Jespers, Eric
%A Río Mateos, Ángel del
%A Ruiz Marín, Manuel
%T Groups generated by two bicyclic units in integral group rings
%D 2002
%@ 1433-5883
%U http://hdl.handle.net/10317/999
%X In [5] Ritter and Sehgal introduced the following units, called the bicylic units, in the unit group
U(ZG) of the integral group ring ZG of a finite group G:
¯a;g = 1 + (1 ¡ g)abg; °a;g = 1 + bga(1 ¡ g);
where a; g 2 G and bg is the sum of all the elements in the cyclic group hgi.
It has been shown that these units generate a large part of the unit group of ZG. Indeed, for
most finite groups G, the bicyclic units together with the Bass cyclic units generate a subgroup of
finite index in U(ZG) [3, 6]. The Bass cyclic units are only needed to cover a subgroup of finite index
in the centre and the group B generated by the bicyclic units contains a subgroup of finite index
in a maximal Z-order of each non-commutative simple image Mn(D) of the rational group algebra
QG. In particular, if n > 1, then B contains a subgroup of finite index in SLn(O), where O is a
maximal order in D; and hence B contains free subgroups of rank two. A next step in determining
the structure of U(ZG) is to investigate relations among the discovered generators. Presently this
is beyond reach. Hence a more realistic goal is to study the structure of the group generated by
two bicyclic units. In [4] Marciniak and Sehgal proved that if ¯a;g is a non trivial unit in ZG (here
G is not necessarily finite) then the group h¯a;g; °a¡1;g¡1i is free of rank 2. Clearly, bicyclic units
are of the form 1 + a with a2 = 0. Salwa, in [7], used the ideas of Marciniak and Sehgal to prove
that if x and y are two elements of an additively torsion-free ring such that x2 = y2 = 0 and xy is
not nilpotent then h(1 + x)m; (1 + y)mi is free of rank 2 for some positive integer m. In particular,
if b1 and b2 are two bicyclic units and (b1 ¡ 1)(b2 ¡ 1) is not nilpotent, then hbm
1 ; bm
2 i is free of
rank 2 for some positive integer m. In this paper we investigate the minimum positive integer m
so that hbm
1 ; bm
2 i is free provided that b1 and b2 are two bicyclic units so that (b1 ¡1)(b2 ¡1) is not
nilpotent. We prove the following theorem which indicates that if b1 and b2 are of the same type
then frequently m = 1.
%K Economía Aplicada
%K Bicylic units
%K Bass cyclic units
%K Structure of the group
%K Dihedral group
%K Unidades bicíclicas
%K Unidades cíclicas de Bass
%K Estructuras de grupo
%K Grupo diédrico
%~ GOEDOC, SUB GOETTINGEN